You run into the trouble of having to defend why your way to fit the divergent series into a pattern is the right one—other approaches may give different results.
The claim is that they don’t: any pattern that points to a finite result points to the same one. If you want proof, then you need a more rigorous formalism.
Now the problem is this pattern leads to a contradiction because it can equally prove any number you want. So we don’t choose to use it as a definition for an infinite sum.
So you need to do a bit more work here to define what you mean here.
You run into the trouble of having to defend why your way to fit the divergent series into a pattern is the right one—other approaches may give different results.
The claim is that they don’t: any pattern that points to a finite result points to the same one. If you want proof, then you need a more rigorous formalism.
Sure, but you’re just claiming that, and I don’t think it’s actually true.
That’s clearly not true in a general sense. Here’s a pattern that points to a different sum:
1 + 2 + 3 + … = 1 + (1 + 1) + (1 + 1 + 1) + … = 1 + 1 + 1 + … = − 1⁄2
Now the problem is this pattern leads to a contradiction because it can equally prove any number you want. So we don’t choose to use it as a definition for an infinite sum.
So you need to do a bit more work here to define what you mean here.